The Mobius ribbon is a surface that has something magical: it has one face, it is not adjustable and if we walk above we can travel all the points of its surface without ever changing the way. If we take one paper striplet’s make a half a lap to one of the two extremes and there we glue togetherwe really get a Mobius ribbon. A figure so simple to build is actually extremely complex. It does not have a “internal” side and an “external” side, it has a single edge and based on how the cut we get different figures. His greatest peculiarity is to be one “Infinite road”: If with a finger we start walking the ribbon and mark our path gradually, we can easily see that we will cover the entire surface of the sheet without ever detaching the finger.
Discovery e studied in 1858 From the German mathematician August Ferdinand Möbius, this figure found applications in the Transportersin electronic circuits, in tricks Of magic And even in the DNA study. In this article we see some of its curious properties and explain the mathematics behind the make -up of the “Afghan bands”.
What is Möbius’ ribbon and how it is built
The Möbius ribbon was discovered and studied independently of two German mathematics: Johann Benedict Listing and August Ferdinand Möbius, from which he takes his name. Building a Mobius ribbon is very simple. Just take a piece of paper rectangular with the longer side equal to at least twice the shortest side – In short, a strip of paper! – make half a lap to one of the extremes and then glue them together. What we will get will be similar to the image.
This structure has a particular geometry. In fact, if we take a pencil and start a trace a line on the entire length without ever detaching the pencil from the sheet, we will be able to return to the starting point having traveled the entire surface Both “inside” and “outside” without ever having to cross the edge. This happens precisely because the Mobius ribbon it does not have one inside and an out! In a classic surface, such as the cylinder, there is one internal and one external side and we can go from one to the other only by crossing one of the two edges, the upper or lower one. If we put the pencil on the external side of the cylinder, in fact, we would return to the starting point without ever having colored the internal part. In the Mobius ribbon, however, There is only one sidewhich constitutes the entire surface, and only one edge. On this surface it cannot be defined uniquely what is inside and what is outbut not even what is below and what is above. In a slightly more formal way, it is said that Möbius’ ribbon is a surface “non -adjustable“.
To better understand what it means, let’s look at the figure:
A mathematical way to represent a cylinder and a möbius ribbon are these diagrams. When we glue the ends with each other, The arrows must match. In the top diagram, which represents a cylinder, we must not make any rotation when we glue the paper, and if we “pass” the face through the point where we have glued the ends, this will reach the other end of the strip while remaining oriented in the same verse and always finding himself up. The cylinder, in fact, is an adjustable surface. In the Mobius ribbon, which is an non -adjustable surface, we see instead that when It will pass through the “seam” pointthe face he will find himself completely overturned.
What happens if we cut the Mobius tape in half: the magic tricks
If we cut a cylinder to halfwe get Two separate ringsbut if we cut the same Mobius ribbonwe get a single ring. This effect is also exploited in magic tricks, as in the game of “Afghan bands”. In this trick, a cylinder, a Mobius tape and a tape to which a complete twist was cut in half before gluing. In the first case, two separate rings are obtained, in the second one ring and in the third two intertwined rings. The latter effect can also be obtained by cutting the Möbius ribbon at one third of the width. To understand why it happens, we use the diagrams again.
If we look at the diagram we see that the upper part of the strip yes connect with the lower part (the two “reds), and then reconnect with the upper part (the two “BL” B “). So, we are getting A single all connected ring. If, on the other hand, we cut it to A third of the widththe upper part will connect to the lower part as in the first case, but now we will have a second ring composed only on the central part. This smaller ring will also be a Möbius ribbon and will be woven with the larger ring. Try to believe.
The applications of the Mobius ribbon
Precisely for its geometric properties, Möbius’ was used in numerous technical applicationsfrom the continuous cycle registration ribbons, to the tapes of the typewriters and in the computer print cartridges. In the past, the Machinery transmission straps They often had half twist, so as to use the entire surface of the ribbon in equal size, lasting longer. The first written testimony of this industrial use is of the 1871but we can find Machinery drawings who already use it in the “Book of knowledge of ingenious mechanical devices” by Al-Jazari del 1206. Even in creative areas, such as the design of Russian mountains and iconic bridges such as the Wuchazi in China, the Mobius ribbon continues to inspire solutions that combine science and design.
